Discovering Statistics Using IBM SPSS Statistics: North American Edition

by Andy Field

the direction of the t-statistic here is similarly influenced by which group we select to be the base category

The Bayesian estimates of the difference between means (i.e., the b-value for group as a predictor of mischief) are in the columns labeled Posterior Mode and Posterior Mean.

With the paired-samples t-test we’re interested in the sampling distribution of the difference scores (not the raw scores).

if you want to test for normality before a paired-samples t-test then you should compute the differences between scores, and then check if this new variable is normally distributed as a proxy for the sampling distribution

The grand mean is the mean of all scores, and for the current data this value will be the mean of all 24 scores. The means we have just calculated are the average score for each participant; if we take the average of those mean scores, we will have the grand mean

Remember that confidence intervals are constructed such that in 95% of samples the intervals contain the true value of the mean difference. So, assuming that this sample’s confidence interval is one of the 95 out of 100 that contain the population value, we can say that the true mean difference lies between −1.83 and −0.67. The importance of this interval is that it does not contain zero (both limits are negative), which tells us that the true value of the mean difference is unlikely to be zero.

SPSS reports the ratio of the null to the alternative hypothesis, so this value means that the data are 0.005 times as probable under the null hypothesis as under the alternative.

Some argue that you need to factor in the between scores in treatment conditions by dividing the estimate of d by the square root of 1 minus the correlation between the scores

you usually state the finding to which the test relates and then report the test statistic, its degrees of freedom and its probability value. This applies whether we use a t-test or a robust test. Ideally report an estimate of the effect size and the Bayes factor too. If you used a robust test you should cite R (R Core Team, 2016) and the WRS2 package (Mair, Schoenbrodt, & Wilcox, 2017), because that’s what was used to compute them.

The explanation is that repeated-measures designs have relatively more power. When the same entities are used across conditions the unsystematic variance (error variance) is reduced dramatically, making it easier to detect the systematic variance. It is often assumed that the way in which you collect data is irrelevant, and in terms of the effect size it sort of is, but if you’re interested in significance then it matters.

this chapter tells us all about the statistical models that we use to analyze situations in which we want to compare more than two independent means.

The model is typically called analysis of variance (or ANOVA to its friends) but, as we shall see, it is just a variant on the linear model.

we include a predictor variable containing two categories into the linear model then the resulting b for that predictor compares the difference between the mean score for the two categories.

if we want to include a categorical predictor that contains more than two categories, this can be achieved by recoding that variable into several categorical predictors each of which has only two categories (dummy coding).

if we’re interested in comparing more than two means we can use the li...

we test the overall fit of a linear model with an F-statistic, we can do the same here: we first use an F to test whether we significantly predict the outcome variable by using group means (which tells us whether, overall, the group means are significantly different) and then use the ...

The ‘ANOVA’ to which some people allude is simply the F-statistic that we encountered as a test of the fit of a linear model, it’s just that the linear model consists of group means.

This approach is fine for simple designs, but becomes impossibly cumbersome in more complex situations such as analysis of covariance or when you have unequal sample sizes.

the linear model extends very logically to the more complex situations (e.g., multiple predictors, unequal group sizes) without the need to get bogged down in mathematics.

The design of this study mimics a very simple randomized controlled trial (as used in pharmacological, medical and psychological intervention trials) because people are randomized into a control group or groups containing the active intervention (in this case puppies, but in other cases a drug or a surgical procedure).

We’ve seen that with two groups we can use a linear model, by replacing the ‘model’ in equation (12.1) with one dummy variable that codes two groups (0 for one group and 1 for the other) and an associated b-value that would represent the difference between the group means

we’ve also seen that this situation is easily incorporated into the linear model by including two dummy variables (each assigned a b-value), and that any number of groups can be included by extending the number of dummy variables to one less than the number of groups

The baseline category should be the condition against which you intend to compare the other groups. In most well-designed experiments there will be a group of participants who act as a control for the other groups and, other things being equal, this will be your baseline category – although the group you choose will depend upon the particular hypotheses you want to test.

When we are predicting an outcome from group membership, predicted values from the model (the value of happiness in equation (12.2)) are the group means.

Using dummy coding is only one of many ways to code dummy variables.

12.1. The overall fit of the model has been tested with an F-statistic (i.e., ANOVA),

Given that our model represents the group means, this F tells us that using group means to predict happiness scores is significantly better than using the mean of all scores: in other words, the group means are significantly different.

The F-test is an overall test that doesn’t identify differences between specific means. However, the model parameters (the b-values) do.

the constant (b0) is equal to the mean of the base category (the control group),

the F-statistic (or F-ratio as it’s also known) tests the overall fit of a linear model to a set of observed data. F is the ratio of how good the model is compared to how bad it is (its error).

The model that represents ‘no effect’ or ‘no relationship between the predictor variable and the outcome’ is one where the predicted value of the outcome is always the grand mean (the mean of the outcome variable).

We can fit a different model to the data that represents our alternative hypotheses.

The intercept and one or more parameters (b) describe the model.

The parameters determine the shape of the model that we have fitted;

In experimental research the parameters (b) represent the differences between group means.

If the differences between group means are large enough, then the resulting model will be a better fit to the data than the null model (grand mean).

our group means are significantly different from the null (that all means are the same).

We use the F-statistic to compare the improvement in fit due to using the model (rather than the null or grand mean, model) to the error that still remains.

the F-statistic is the ratio of the explained to the unexplained variation. We calculate this var...

To find the total amount of variation within our data we calculate the difference between each observed data point and the grand mean. We square these differences and add them to give us the total sum of squares (SST): (12.9)

The variance and the sums of squares are related such that variance, s2 = SS/(N − 1), where N is the number of observations

we can calculate the total sum of squares from the variance of all observations (the grand variance) by rearranging t...

grand variance is the variation between all scores, regardless of the group fro...

vary. For SST, we used the entire sample (i.e., 15 scores) to calculate the sums of squares and so the total degrees of freedom (dfT) are one less than the total sample size (N − 1).

Because our model predicts the outcome from the means of our treatment (puppy therapy) groups, the model sums of squares tell us how much of the total variation in the outcome can be explained by the fact that different scores come from entities in different treatment conditions.

Given that the predicted value for participants in a group is the same value (the group mean), the easiest way to calculate SSM is by using: (12.11)

This equation basically says: Calculate the difference between the mean of each group and the grand mean . Square each of these differences. Multiply each result by the number of participants within that group (ng). Add the values for each group together.

For SSM, the degrees of freedom (dfM) are one less than the number of ‘things’ used to calculate the SS. We used the three group means, so dfM is the number of groups minus one (which you’ll see denoted as k − 1).

The simplest way to calculate SSR is to subtract SSM from SST (SSR = SST − SSM), but this provides little insight into what SSR represents and, of course, if you’ve messed up the calculations of either SSM or SST (or both!) then SSR will be incorrect also.

SSR is calculated by looking at the difference between the score obtained by a person and the mean of the group to which the person belongs.